Abstract
This articles starts with a study of the Machina triangle drawn based on upper-moments of distributions. This allows to establish clearly that one should be cautious about the choice of renormalized or non renormalized upper moments. Then, the concepts of prudence and temperance are revisited - and new coefficients introduced - based on a development up to order 4 of the risk premium. This article also shows how prudence and temperance type coefficients can be computed under a HARA utility hypothesis. An illustration is then conducted on monoperiodic portfolio optimization. Remarkably, a closed form formula is obtained for the optimal weight of a portfolio consisting of a risk-free and a risky asset and managed looking at the moments up to order 4of the risky asset.